This paper is an impressive contribution to the theory of \(p\)-local finite groups with a number of very interesting results, including the following two theorems: Theorem A: Let \(p\) be an odd prime. If \(X\) is a \(1\)-connected \(p\)-compact group, \(q\) a prime power prime to \(p\), and \(\tau\) an automorphism of \(X\) of finite order prime to \(p\), then the space of homotopy fixed points of \(BX\) by the action of \(\tau\psi^q\) is the classifying space of a \(p\)-local finite group. Theorem B: Let \(X\) be a connected \(p\)-compact group. If \(G\) is a finite group of order prime to \(p\) and \(\rho : G \to \mathrm{Out}(X)\) and outer action, then the following hold: (1) \(\rho\) lifts to a unique action of \(G\) on \(X\), up to equivalence; (2) \(X^{hG}\) is a connected \(p\)-compact group with \(H^*(BX^{hG};\mathbb{Q}_p) \cong S[QH^*(BX;\mathbb{Q}_p)_G]\), the symmetric algebra generated on the coinvariants \(QH^*(BX;\mathbb{Q}_p)_G\); (3) \(X^{hG} \to X\) is a \(p\)-compact group monomorphism, there is a homotopy equivalence \(X \simeq X^{hG} \times X/X^{hG}\) and \(X/X^{hG}\) is an \(H\)-space; (4) If \(p\) is odd and \(H^*(BX;\mathbb{F}_p)\) is a polynomial ring, then \(H^*(BX^{hG};\mathbb{F}_p)\) is also a polynomial ring.